- Research Article
- Open Access
Global Optimal Regularity for the Parabolic Polyharmonic Equations
© Fengping Yao. 2010
- Received: 21 February 2010
- Accepted: 3 June 2010
- Published: 14 June 2010
- Orlicz Space
- Young Function
- Regularity Estimate
- Interior Regularity
- Covering Lemma
Regularity theory in PDE plays an important role in the development of second-order elliptic and parabolic equations. Classical regularity estimates for elliptic and parabolic equations consist of Schauder estimates, estimates, De Giorgi-Nash estimates, Krylov-Safonov estimates, and so on. and Schauder estimates, which play important roles in the theory of partial differential equations, are two fundamental estimates for elliptic and parabolic equations and the basis for the existence, uniqueness, and regularity of solutions.
where , and is a positive integer. Since the 1960s, the need to use wider spaces of functions than Sobolev spaces arose out of various practical problems. Orlicz spaces have been studied as the generalization of Sobolev spaces since they were introduced by Orlicz  (see [2–6]). The theory of Orlicz spaces plays a crucial role in many fields of mathematics (see ).
where is a constant independent from and . Indeed, if with , (1.8) is reduced to classical estimates. We remark that although we use similar functional framework and iteration-covering procedure as in [6, 12], more complicated analysis should be carefully carried out with a proper dilation of the unbounded domain.
Here for the reader's convenience, we will give some definitions on the general Orlicz spaces.
Lemma 1.7 (see ).
Now let us state the main results of this work.
In this section, we will finish the proof of the main result, Theorem 1.9. Just as , we will use the following two lemmas. The first lemma is the following integral inequality.
Lemma 3.1 (see ).
Moreover, we recall the following result.
Lemma 3.2 (see [10, Theorem ]).
Thus we can obtain the desired result (3.10).
Now we are ready to prove the main result, Theorem 1.9.
The author wishes to thank the anonymous referee for offering valuable suggestions to improve the expressions. This work is supported in part by Tianyuan Foundation (10926084) and Research Fund for the Doctoral Program of Higher Education of China (20093108120003). Moreover, the author wishes to thank the department of mathematics at Shanghai university which was supported by the Shanghai Leading Academic Discipline Project (J50101) and Key Disciplines of Shanghai Municipality (S30104).
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